Simplify the following expression and state the condition under which the simplification is valid. You can assume that $t \neq 0$. $p = \dfrac{5}{4t - 40} \div \dfrac{3}{7t^2 - 70t} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{5}{4t - 40} \times \dfrac{7t^2 - 70t}{3} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ 5 \times (7t^2 - 70t) } { (4t - 40) \times 3 } $ $ p = \dfrac {5 \times 7t(t - 10)} {3 \times 4(t - 10)} $ $ p = \dfrac{35t(t - 10)}{12(t - 10)} $ We can cancel the $t - 10$ so long as $t - 10 \neq 0$ Therefore $t \neq 10$ $p = \dfrac{35t \cancel{(t - 10})}{12 \cancel{(t - 10)}} = \dfrac{35t}{12} $